The asymptotic winding at a negative puncture is one more than the winding of $\pi\partial_s u$ along small circles

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level, $J$ a $d\lambda_0$-compatible complex structure on $\xi$, and $\tilde J$ the induced almost complex structure on $\mathbb{R}^4\setminus\{0\}\cong\mathbb{R}\times S$. Let $U\subset\mathbb{C}$ be open and contain the punctured disc $B(z_0,r)\setminus\{z_0\}$, $r>0$, and let $u$ be smooth, nonvanishing and $\tilde J$-holomorphic on $U$. Suppose that the $\xi$-component $\pi\,\partial_s u$, read in Hryniewicz's global frame of $\xi$, has winding number $k$ along every counterclockwise circle $t\mapsto z_0+\rho e^{2\pi it}$ with $0<\rho<r$. Then at $z_0$ seen as a negative puncture, in the cylindrical coordinates $(s,t)\mapsto z_0+e^{-2\pi(s+it)}$ with $t$ traversed backwards (sign $-1$),

This expresses the asymptotic winding at a negative puncture, which is defined in cylindrical coordinates, through the winding of $\pi\partial_s u$ along small counterclockwise circles about $z_0$ in the plane coordinate.

Formalization Note $\mathbb{R}^4$ is V 1, $\tilde J$ is Jtilde H J, and $\pi\partial_s u$ in the global frame is piDs H u : ℂ → ℂ. The circles are z₀ + ccwLoop ρ t. The conclusion is HasWindInfAt H u (cylEndAt z₀) (-1) (k + 1): for all large $s$ the loop $t\mapsto\pi\partial_s w(s,-t)$ of $w=\mathrm{pr}\circ u\circ\mathrm{cylEndAt}\,z_0$ has winding number $k+1$. No energy bound or asymptotic hypothesis is assumed.